It is also necessary to rejoice on Sabbaths, Festivals and on Purim because all of them are a memorial to our exodus from Egypt and of the miracles which He has done wondrously with His chosen ones.

My Question:

What does Purim have to do with remembering the Exodus from Egypt1?

1. While it can also be asked "What does the Sabbath have to do with remembering the Exodus?" or "What do the festivals have to do with remembering the Exodus?" the connection between Purim and remembering the Exodus is what intrigues the author of this question.

3 Answers
3

In the commentary of Alei Orach on Orchos Tzadikim, he suggests:
1 - "כולם" is referring to all but Purim.
2 - On Purim we accepted the Torah voluntarily, "קימו וקיבלו היהודים" which was the ultimate goal of Yitzias Mitzrayim.

Haman is a descendent from Amalek (עמלק), the fruit of Amalek. Amalek is from the spiritual root of the Nachash found in Bereshit. This is based upon the prophecy of Isaiah 14:29 which says:

אל־תשמחי פלשת כלך כי נשבר שבט מכך כי־משרש נחש יצא צפע ופריו שרף מעופף

That צפע is gematria Amalek. And on the side of holiness, Mordechai is the gilgul of Moshe Rabbeinu. Rabbi Yolles points out that the Roshei Teivot of Haman (המן) is המן משורש נחש, that Haman, like his ancestor Amalek, is from the root of the Nachash.

It’s interesting to note from his full name in Megillat Esther (המן בן המדתא האגגי) that the Mispar Katan with the 4 words of his name (a sum of 56) equals the Mispar Katan of שרף מעופף when including the 8 letters and 2 words. That complies with both details in the posuk from Isaiah, that Amalek is from the root of the Nachash and Haman is (Amalek’s fruit) his descendent.

And all this concept that Purim is in memory of the Exodus from Egypt follows the idea presented in Shavuot 39a that Purim is when the Jewish people fulfilled that which they accepted at the giving of the Torah. Like it says:

There is a discussion in the Ramban at the end of Parshas Bo (Shmos 13:16) explaining why the Torah connected many Mitzvos with יציאת מצרים.
He explains the principle of miracles done for the benefit of Am Yisrael, and their connection with יציאת מצרים.